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On what I do not understand (and have something to say), model theory
 Mathematica Japonica, submitted. [Sh:702]; math.LO/9910158
"... Abstract. This is a nonstandard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried ..."
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Cited by 23 (8 self)
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Abstract. This is a nonstandard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments. (666) revision:20011112 modified:20031118
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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Cited by 4 (3 self)
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
Some Aspects and Examples of Infinity Notions
, 1994
"... Our main contribution is a formal definition of what could be called a Tnotion of infinity, for set theories T extending ZF. Around this definition we organize some old and new notions of infinity; we also indicate some easy independence proofs. ..."
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Cited by 1 (0 self)
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Our main contribution is a formal definition of what could be called a Tnotion of infinity, for set theories T extending ZF. Around this definition we organize some old and new notions of infinity; we also indicate some easy independence proofs.
SINGULAR CARDINALS: FROM HAUSDORFF’S GAPS TO SHELAH’S PCF THEORY
"... The mathematical subject of singular cardinals is young and many of the mathematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discu ..."
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The mathematical subject of singular cardinals is young and many of the mathematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discussions with some of the people who created the 20th century history of singular cardinals made the writing of this article fascinating. I am indebted to Moti Gitik, Ronald Jensen, István Juhász, Menachem Magidor and Saharon Shelah for the time and effort they spent on helping me understand the development of the subject and for many illuminations they provided. A lot of what I thought about the history of singular cardinals had to change as a result of these discussions. Special thanks are due to István Juhász, for his patient reading for me from the Russian text of Alexandrov and Urysohn’s Memoirs, to Salma Kuhlmann, who directed me to the definition of singular cardinals in Hausdorff’s writing, and to Stefan Geschke, who helped me with the German texts I needed to read and
MSc in Logic
, 2011
"... Degrees of nondeterminacy and game logics on cardinals under the axiom of determinacy ..."
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Degrees of nondeterminacy and game logics on cardinals under the axiom of determinacy
Set theory without choice: . . .
, 2003
"... We prove in ZF+DC, e.g. that: if µ = H(µ)  and µ> cf(µ)> ℵ0 then µ + is regular but non measurable. This is in contrast with the results on measurability for µ = ℵω due to Apter and Magidor [ApMg]. ..."
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We prove in ZF+DC, e.g. that: if µ = H(µ)  and µ> cf(µ)> ℵ0 then µ + is regular but non measurable. This is in contrast with the results on measurability for µ = ℵω due to Apter and Magidor [ApMg].
Set theory without choice: Not . . .
, 1995
"... We prove in ZF+DC, e.g. that: if µ = H(µ)  and µ> cf(µ)> ℵ0 then µ + is regular but non measurable. This is in contrast with the results on measurability for µ = ℵω due to Apter and Magidor [ApMg]. ..."
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We prove in ZF+DC, e.g. that: if µ = H(µ)  and µ> cf(µ)> ℵ0 then µ + is regular but non measurable. This is in contrast with the results on measurability for µ = ℵω due to Apter and Magidor [ApMg].
REGULAR CARDINALS IN MODELS OF ZF BY
"... Abstract. We prove the following Theorem. Suppose M is a countable model of ZFC and k is an almost huge cardinal in M. Let A be a subset of k consisting of nonlimit ordinals. Then there is a model NA of ZF such that S0 is a regular cardinal in NA iff a e A for every a> 0. 0. Introduction. We conside ..."
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Abstract. We prove the following Theorem. Suppose M is a countable model of ZFC and k is an almost huge cardinal in M. Let A be a subset of k consisting of nonlimit ordinals. Then there is a model NA of ZF such that S0 is a regular cardinal in NA iff a e A for every a> 0. 0. Introduction. We consider the following question. What are the restrictions in ZF on the class of all regular cardinals? Clearly, S0 is always regular and Nu, Sw+l0, SuU and Sa, for a the least s.t. Sa = a, are singular. For a limit a, if Na is regular, then it is already quite large and its existence is unprovable in ZF. In ZFC, Na+1 is a regular cardinal for every a. Feferman and Levy [3] proved that it is not true in ZF
Successor Large Cardinals in Symmetric Extensions ∗
"... We give an exposition in modern language (and using partial orders) of Jech’s method for obtaining models where successor cardinals have large cardinal properties. In such models, the axiom of choice must necessarily fail. In particular, we show how, given any regular cardinal and a large cardinal o ..."
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We give an exposition in modern language (and using partial orders) of Jech’s method for obtaining models where successor cardinals have large cardinal properties. In such models, the axiom of choice must necessarily fail. In particular, we show how, given any regular cardinal and a large cardinal of the requisite type above it, there is a symmetric extension of the universe in which the axiom of choice fails, the smaller cardinal is preserved, and its successor cardinal is measurable, strongly compact or supercompact, depending on what we started with. The main novelty of the exposition is a slightly more general form of the LévySolovay Theorem, as well as a proof that fine measures generate fine measures in generic extensions obtained by small forcing. 1